Abstract
The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.
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References
Arfken G.B., Weber H.-J.: Mathematical Methods for Physicists, 6th ed. Elsevier, San Diego (2005)
Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957)
Eshelby J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569 (1959)
Gao X.-L.: A mathematical analysis of the elasto-plastic anti-plane shear problem of a power-law material and one class of closed-form solutions. Int. J. Solids Struct. 33, 2213–2223 (1996)
Gao X.-L., Li K.: A shear-lag model for carbon nanotube-reinforced polymer composites. Int. J. Solids Struct. 42, 1649–1667 (2005)
Gao X.-L., Liu M.Q.: Strain gradient solution for the Eshelby-type polyhedral inclusion problem. J. Mech. Phys. Solids 60, 261–276 (2012)
Gao X.-L., Ma H.M.: Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mech. 207, 163–181 (2009)
Gao X.-L., Ma H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)
Gao X.-L., Ma H.M.: Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. A 466, 2425–2446 (2010)
Gao X.-L., Ma H.M.: Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta. Mech. 223, 1067–1080 (2012)
Gao X.-L., Park S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)
Gao X.-L., Zhou S.-S.: Strain gradient solutions of half-space and half-plane contact problems. Z. Angew. Math. Phys. 64, 1363–1386 (2013)
Horgan C.O.: Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995)
Kawashita M., Nozaki H.: Eshelby tensor of a polygonal inclusion and its special properties. J. Elast. 64, 71–84 (2001)
Le Quang H., He Q.-C., Zheng Q.-S.: Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity. Int. J. Solids Struct. 45, 3845–3857 (2008)
Liu L.P.: Solutions to the Eshelby conjectures. Proc. R. Soc. A 464, 573–594 (2008)
Liu M.Q., Gao X.-L.: Strain gradient solution for the Eshelby-type polygonal inclusion problem. Int. J. Solids Struct. 50, 328–338 (2013)
Lubarda V.A.: Circular inclusions in anti-plane strain couple stress elasticity. Int. J. Solids Struct. 40, 3827–3851 (2003)
Ma H.M., Gao X.-L.: Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory. Acta Mech. 211, 115–129 (2010)
Ma H.M., Gao X.-L.: Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix. Int. J. Solids Struct. 48, 44–55 (2011)
Ma, H.M., Gao, X.-L.: Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem. Int. J. Solids Struct. (in press) (Published online in August 2013) (doi:10.1016/j.ijsolstr.2013.07.011) (2013a)
Ma, H.M., Gao, X.-L.: A homogenization method based on a simplified strain gradient elasticity theory. Acta Mech. (in review) (Submitted in July 2013) (2013b)
Mindlin R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)
Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Nozaki H., Taya M.: Elastic fields in a polygon-shaped inclusion with uniform eigenstrains. ASME J. Appl. Mech. 64, 495–502 (1997)
Rodin G.J.: Eshelby’s inclusion problem for polygons and polyhedra. J. Mech. Phys. Solids. 44, 1977–1995 (1996)
Sadd M.H.: Elasticity: Theory, Applications, and Numerics, 2nd edition. Academic Press, Burlington MA, USA (2009)
Waldvogel J.: The Newtonian potential of homogeneous polyhedra. Z. Angew. Math. Phys. 30, 388–398 (1979)
Weng G.J.: The theoretical connection between Mori-Tanaka’s theory and the Hashin-Shtrikman-Walpole bounds. Int. J. Eng. Sci. 28, 1111–1120 (1990)
Xu B.X., Wang M.Z.: Special properties of Eshelby tensor for a regular polygonal inclusion. Acta. Mech. Sinica. 21, 267–271 (2005)
Xu B.X., Wang M.Z.: The arithmetic mean theorem for the N-fold rotational symmetrical inclusion in anti-plane elasticity. Acta Mech. 194, 233–242 (2007)
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Liu, M.Q., Gao, XL. Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory. Acta Mech 225, 809–823 (2014). https://doi.org/10.1007/s00707-013-0991-2
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DOI: https://doi.org/10.1007/s00707-013-0991-2